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MEASURES OF PSEUDORANDOMNESS FOR FINITE SEQUENCES: TYPICAL VALUES
 

Summary: MEASURES OF PSEUDORANDOMNESS FOR FINITE
SEQUENCES: TYPICAL VALUES
N. ALON, Y. KOHAYAKAWA, C. MAUDUIT, C. G. MOREIRA, AND V. R¨ODL
Abstract. Mauduit and S´ark¨ozy introduced and studied certain nu-
merical parameters associated to finite binary sequences EN {-1, 1}N
in order to measure their `level of randomness'. Those parameters, the
normality measure N(EN ), the well-distribution measure W(EN ), and
the correlation measure Ck(EN ) of order k, focus on different combinato-
rial aspects of EN . In their work, amongst others, Mauduit and S´ark¨ozy
(i) investigated the relationship among those parameters and their min-
imal possible value, (ii) estimated N(EN ), W(EN ), and Ck(EN ) for
certain explicitly constructed sequences EN suggested to have a `pseu-
dorandom nature', and (iii) investigated the value of those parameters
for genuinely random sequences EN .
In this paper, we continue the work in the direction of (iii) above
and determine the order of magnitude of N(EN ), W(EN ), and Ck(EN )
for typical EN . We prove that, for most EN {-1, 1}N
, both W(EN )
and N(EN ) are of order

  

Source: Alon, Noga - School of Mathematical Sciences, Tel Aviv University
Kohayakawa, Yoshiharu - Departamento de Ciência da Computação, Universidade de São Paulo
Moreira, Carlos Gustavo - Instituto Nacional de Matemática Pura e Aplicada (IMPA)

 

Collections: Computer Technologies and Information Sciences; Mathematics